Unit 8.4 | Bernoulli’s Equation
- 17 minutes
- 1 formulas
- 22 questions
- 4 videos
Faster fluid, lower pressure. Bernoulli’s equation is conservation of energy for flowing fluids, and it explains wings, trains and draining tanks.
Unit 8 Breakdown
In this lesson:
- Bernoulli’s equation as conservation of energy for a flowing fluid
- Why faster fluid has lower pressure, and where that shows up: wings, trains, shower curtains
- Torricelli’s theorem for a draining tank: [katex] v = \sqrt{2gh} [/katex]
- The 4 problem types: fast flow and low pressure, horizontal pipes, pipes that change height, and draining tanks
Introduction
This lesson covers Bernoulli’s equation, the last big idea of AP Physics 1. It ties fluid pressure, speed and height together, and it is just conservation of energy in a new form.
In Unit 8.3 the water sped up in a narrow pipe. Faster water has more kinetic energy, and that energy has to come from somewhere. It comes from the pressure: the fluid is pushed from a region of high pressure into a region of low pressure, and that push does work on it.
Bernoulli’s Equation
Follow a small parcel of fluid of volume [katex] V [/katex] along the pipe. Use conservation of energy, with every term divided by the volume:
- Kinetic energy per volume: [katex] \frac{\tfrac{1}{2}mv^2}{V} = \tfrac{1}{2}\rho v^2 [/katex].
- Gravitational potential energy per volume: [katex] \frac{mgy}{V} = \rho g y [/katex].
- The surrounding fluid pushes the parcel from high pressure to low. The work it does, per volume, is the pressure difference [katex] P_1 – P_2 [/katex].
[katex] P_1 + \rho g y_1 + \tfrac{1}{2}\rho v_1^2 = P_2 + \rho g y_2 + \tfrac{1}{2}\rho v_2^2 [/katex]
Along an ideal fluid’s flow, the sum of pressure, [katex] \rho g y [/katex] and [katex] \tfrac{1}{2}\rho v^2 [/katex] stays the same. Raise one term and another must drop. If the fluid is not moving ([katex] v = 0 [/katex]), it gives back [katex] P = P_0 + \rho g h [/katex] from Unit 8.1.
Bernoulli’s equation is conservation of energy, and the continuity equation is conservation of mass. Most flow problems use both.
▶ Watch: Bernoulli’s equation, worked from scratch (starts at 20:46).
Every Bernoulli question on the AP exam is one of the four types below.
| Problem type | The clue | The setup |
| 1. Fast flow, low pressure | wings, trains, curtains, “which point has the lower pressure” | Same height: faster means lower [katex] P [/katex] |
| 2. Horizontal pipes | a level pipe that narrows, a Venturi tube | Continuity for [katex] v_2 [/katex], then [katex] P_1 – P_2 = \tfrac{1}{2}\rho(v_2^2 – v_1^2) [/katex] |
| 3. Pipes that change height | “on the second floor”, “a height h higher” | Full Bernoulli, all three terms |
| 4. Draining tanks | a hole in a tank, a spigot, “efflux speed” | [katex] v = \sqrt{2gh} [/katex], then projectile motion |
Problem Type 1: Fast flow, low pressure
How to spot it: No numbers, just “why”: airplane wings, a train passing a platform, a shower curtain, a roof in a storm, air blown between two papers. Or “which point has the lower pressure”.
Setup: At the same height, [katex] P + \tfrac{1}{2}\rho v^2 [/katex] is constant: where the fluid is faster, its pressure is lower. Find where the flow is fastest, then ask which way the higher pressure pushes.
A train speeds past a person standing on the platform. Is the person pushed toward the train or away from it?
The air between the person and the train is dragged along and moves fast, so its pressure drops. The still air on the person’s other side is at normal pressure. The pressure difference pushes the person toward the train. Wings work the same way: faster air over the top, lower pressure there, a net push up.
▶ Watch: the conceptual Bernoulli examples (starts at 24:59).
AP trap: Thinking the narrow section has higher pressure because the fluid is “squeezed”. The fluid speeds up there, and its pressure goes down.
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Problem Type 2: Horizontal pipes: pressure from speed
How to spot it: A level pipe or Venturi tube changes size, and the question asks for a pressure, a pressure difference or a speed.
Setup: Use continuity to get the second speed, then Bernoulli without the height terms: [katex] P_1 – P_2 = \tfrac{1}{2}\rho\,(v_2^2 – v_1^2) [/katex].
Water flows at [katex] 2.0\,\text{m/s} [/katex] through a level pipe with a cross-sectional area of [katex] 10\,\text{cm}^2 [/katex], at a pressure of [katex] 1.8 \times 10^5 \text{ Pa} [/katex]. The pipe narrows to [katex] 2.5\,\text{cm}^2 [/katex]. What is the pressure in the narrow section?
Continuity: the area drops by a factor of 4, so [katex] v_2 = 4(2.0) = 8.0 \text{ m/s} [/katex]. Bernoulli: [katex] P_1 – P_2 = \tfrac{1}{2}(1000)(8.0^2 – 2.0^2) = 3.0 \times 10^4 \text{ Pa} [/katex], so [katex] P_2 = 1.8 \times 10^5 – 0.30 \times 10^5 = 1.5 \times 10^5 \text{ Pa} [/katex].
AP trap: Writing [katex] \tfrac{1}{2}\rho(v_2 – v_1)^2 [/katex]. Square each speed first, then subtract: [katex] 8^2 – 2^2 = 60 [/katex], not [katex] (8 – 2)^2 = 36 [/katex].
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Problem Type 3: Pipes that change height
How to spot it: The fluid rises or falls along the pipe: “pumped up to the second floor”, “a point a height h higher”, a hose up a hill.
Setup: Keep all three terms. Pick the lower point as [katex] y = 0 [/katex]. Use continuity for the speeds. Rising costs pressure ([katex] \rho g\,\Delta y [/katex]), and so does speeding up.
Water enters a house through a basement pipe at [katex] 2.0\,\text{m/s} [/katex] and an absolute pressure of [katex] 3.0 \times 10^5 \text{ Pa} [/katex]. A pipe on a floor [katex] 5.0\,\text{m} [/katex] higher has half the cross-sectional area. Find the pressure there.
Continuity: [katex] v_2 = 2(2.0) = 4.0 \text{ m/s} [/katex]. Bernoulli: [katex] P_2 = P_1 – \tfrac{1}{2}\rho(v_2^2 – v_1^2) – \rho g\,\Delta y [/katex] [katex] = 3.0 \times 10^5 – \tfrac{1}{2}(1000)(16 – 4) – (1000)(9.8)(5.0) [/katex] [katex] = 3.0 \times 10^5 – 6.0 \times 10^3 – 4.9 \times 10^4 \approx 2.45 \times 10^5 \text{ Pa} [/katex]. The climb costs far more pressure than the speed-up.
AP trap: Dropping the height term because the pipe “looks level”, or giving it the wrong sign. Going up, the pressure drops by [katex] \rho g\,\Delta y [/katex]; going down, it rises.
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Problem Type 4: Draining tanks: Torricelli’s theorem
How to spot it: Water leaves a hole or spigot in an open tank: “efflux speed”, “how fast does it leave”, “how far does the stream land”.
Setup: Point 1 at the surface, point 2 at the hole. Both are open to the air ([katex] P_1 = P_2 = P_0 [/katex]), and a wide tank’s surface barely moves ([katex] v_1 \approx 0 [/katex]). Bernoulli leaves [katex] v = \sqrt{2gh} [/katex], with [katex] h [/katex] the depth of the hole below the surface.
Deeper hole, faster stream
An open tank has a small hole in its side, [katex] 1.8\,\text{m} [/katex] below the water surface and [katex] 1.25\,\text{m} [/katex] above the ground. How fast does the water leave, and how far from the tank does it land?
Torricelli: [katex] v = \sqrt{2(9.8)(1.8)} \approx 5.9 \text{ m/s} [/katex], horizontally. Fall time from [katex] 1.25\,\text{m} [/katex]: [katex] t = \sqrt{\frac{2(1.25)}{9.8}} \approx 0.51 \text{ s} [/katex]. Distance: [katex] x = (5.94)(0.505) \approx 3.0 \text{ m} [/katex].
▶ Watch: after it leaves the hole the stream is a horizontal projectile. Here is the projectile method, step by step.
AP trap: Measuring [katex] h [/katex] from the bottom of the tank, or from the ground. It is the depth of the hole below the water surface. The exit speed also does not depend on the size of the hole, the width of the tank or the liquid’s density.
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Accounts are free! Log in to try this question, see explanations, save progress, and more!
Unit 8 Wrap-Up
- Density and pressure: [katex] \rho = \frac{m}{V} [/katex], [katex] P = \frac{F_\perp}{A} [/katex], and pressure grows with depth, [katex] P = P_0 + \rho g h [/katex] (8.1).
- Buoyancy: the fluid pushes up with the weight of the fluid displaced, [katex] F_B = \rho_f V_{sub} g [/katex] (8.2).
- Continuity: what flows in flows out, [katex] A_1 v_1 = A_2 v_2 [/katex] (8.3).
- Bernoulli: energy per volume is conserved, so faster flow means lower pressure (8.4).
Test the whole unit in one go:
You Finished AP Physics 1
Unit 8 is the last unit of the course. You now have every tool the AP exam uses: kinematics, forces, energy, momentum, rotation, oscillations and fluids. What is left is practice, mixing the units the way the exam does.
- All the free notes, worksheets and practice in one place: the AP Physics 1 course materials page.
- One page per unit when you review: Kinematics, Forces, Circular motion, Energy, work and power, Momentum, Torque and rotational kinematics, Rotational energy and angular momentum, Simple harmonic motion, Fluids.
Review the unit you feel least sure of first, then work mixed practice under time.
Accounts are free! Log in to read the rest, try the practice questions, and save your progress.
This article was written by Jason